F70CF Continuous-Time Finance

Dr Torsten KleinowDr Alistair Wallis

Course co-ordinator(s): Dr Torsten Kleinow (Edinburgh), Dr Alistair Wallis (Malaysia).


This course develops the theory and practice of financial derivatives pricing in continuous time. The course contributes towards exemption from the CT8 examination of the Faculty and Institute of Actuaries


We shall introduce the Black Scholes model, and look at some simple extensions to treat dividends/currencies.

We will look at bond pricing and the term structure of interest rates in an arbitrage-free framework. We will also cover how to price bond derivatives, and look at models of credit risk.

This course will concentrate on the application of stochastic modelling, however it will also require some of the basics of the theory of stochastic process, in particular a knowledge of Brownian motion and its properties.

Detailed Information

Course Description: Link to Official Course Descriptor.

Pre-requisite course(s): F79SP Stochastic Processes & F79DF Derivative Markets and Discrete-time Finance .

Location: Edinburgh.

Semester: 1.


  • Theory of Martingales in continuous time
  • Brownian motion; definitions and properties
  • Brownian motion as the limit of a binomial random-walk process
  • Introduction to stochastic integration, stochastic differential equations and Ito's formula
  • Geometric Brownian motion; the Ornstein-Uhlenbeck process
  • Introduction to Girsanov’s theorem and the martingale representation theorem
  • The Black-Scholes model
  • Derivatives pricing using the Black-Scholes model using the martingale and PDE approaches to pricing
  • Extensions to foreign currencies and dividend-paying stocks
  • Portfolio risk management using the Greeks
  • Introduction to interest rate models
  • Introduction to credit risk models

Learning Outcomes: Subject Mastery

At the end of studying this course, students should be able to:

  • Demonstrate a knowledge of Brownian motion and its properties;
  • Show how to calibrate the Binomial model as an approximation to Brownian motion using empirical data;
  • Apply Ito’s Formula, the Girsanov theorem and the martingale representation theorem;
  • Work with martingale measures, and understand their connection with arbitrage free/complete markets;
  • Understand the concepts of replication, hedging, and delta hedging in continuous time;
  • Derive the Black-Scholes formula and the Black-Scholes PDE;
  • Price contingent claims (in particular European style options and forward contracts);
  • Extend the Black-Scholes formula to foreign currencies and dividend paying stocks;
  • Understand the role of the Greeks in portfolio risk management;
  • Derive relationships between forward interest rates, spot rates and zero coupon bond prices;
  • Understand issues involved in selecting and using short rate models for pricing bonds and bond derivatives;
  • Manipulate explicit bond price formulae for the Vasicek and CIR models, and derive the implied forward rate curves;
  • Define the different approaches to modelling credit risk;
  • Define and apply the Merton model for credit risk to price simple corporate bonds and calculate credit spreads;
  • Define and apply the 2-state model for credit risk with deterministic and stochastic transition intensities.

Reading list:

The following books are recommended:

  • Hull, J., (2000) Options, futures and other derivative securities, 4th ed, Prentice Hall;
  • Baxter, M. and Rennie, A. (1996), Financial calculus, Cambridge University Press;

Students may also find the following books useful:

  • Luenberger, D.G. (1998) Investment science. Oxford University Press.
  • Bingham, N. H. & Kiesel, R. (1998) Risk neutral valuation. Pricing and hedging of financial derivatives. Springer Verlag;
  • Björk, T. (1998) Arbitrage theory in continuous time. Oxford University Press;

Assessment Methods: Due to covid, assessment methods for Academic Year 2021/22 may vary from those noted on the official course descriptor. Please see:
- Maths (F1) Course Weightings 2021/22
- Computer Science (F2) Course Weightings 2021/22
- AMS (F7) Course Weightings 2021/22

SCQF Level: 10.

Credits: 15.

Other Information

Help: If you have any problems or questions regarding the course, you are encouraged to contact the course leader.

Canvas: further information and course materials are available on Canvas